AI 中文总结
研究 0-1 规划的 Shor 松弛的奇异性程度，确定线性约束下其至多为 1，通过求解 LP 可恢复 Slater 条件，而二次约束下奇异性程度可能随矩阵阶数线性增长。
AI 中文摘要
Shor 松弛是非凸二次优化的标准工具，但可能不满足 Slater 条件。奇异性程度通过达到最小面所需的面缩减步骤数来量化这种退化。我们确定定义二元可行集的约束如何影响这个量。对于由线性方程和不等式定义的每个非空二元集，Shor 松弛的奇异性程度至多为 1。当线性规划（LP）松弛在开立方体内包含一个严格满足每个不等式的点时，奇异性程度恰好为 0，否则为 1。作为此特征的算法结果，可通过求解 LP 而非辅助半定规划（SDP）来恢复 Slater 条件。与之形成鲜明对比的是，对于每个\(n\geq1\)，我们构造一个由二次等式定义的二元可行集，其 Shor 松弛具有阶数为\(n + 1\)的正半定矩阵变量且奇异性程度为\(n\)。因此，在受线性约束的情况下奇异性程度是一致有界的，但当使用二次定义约束时，它可能随矩阵阶数线性增长。
英文摘要
Shor relaxations are standard tools for nonconvex quadratic optimization, but they may fail Slater's condition. The singularity degree quantifies this degeneracy by the number of facial-reduction steps needed to reach the minimal face. We determine how the constraints defining a binary feasible set affect this quantity. For every nonempty binary set defined by linear equations and inequalities, the Shor relaxation has singularity degree at most one. It is zero precisely when the linear programming (LP) relaxation contains a point in the open cube that strictly satisfies every inequality, and is one otherwise. As an algorithmic consequence of this characterization, Slater's condition can be restored by solving an LP rather than an auxiliary semidefinite program (SDP). In sharp contrast, for every $n\geq1$, we construct a binary feasible set defined by quadratic equalities whose Shor relaxation has a positive semidefinite matrix variable of order $n+1$ and singularity degree $n$. Thus, singularity degree is uniformly bounded in the linearly constrained case but can grow linearly with the matrix order when quadratic defining constraints are used.
